Lie's reduction method and differential Galois theory in the complex analytic context

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February 2012, 32(2): 353-379. doi: 10.3934/dcds.2012.32.353

Lie's reduction method and differential Galois theory in the complex analytic context

David Blázquez-Sanz ^1, and Juan J. Morales-Ruiz ^2,

1.	Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia
2.	Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain

Received December 2010 Revised April 2011 Published September 2011

Abstract
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This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.

Keywords: differential equations in the complex domain., Differential Galois theory.

Mathematics Subject Classification: Primary: 34M05; Secondary: 12H05, 14L99, 34A2.

Citation: David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

References:

[1]	C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474.
[2]	C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649. doi: 10.1088/0305-4470/30/13/015.
[3]	C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614. doi: 10.1088/0305-4470/31/31/008.
[4]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010.
[5]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517.
[6]	R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991.
[7]	A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986.
[8]	J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000.
[9]	J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258.
[10]	J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87. doi: 10.1023/A:1010743114995.
[11]	J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp.
[12]	J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69. doi: 10.1023/A:1013913930134.
[13]	G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779.
[14]	G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.
[15]	G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519. doi: 10.4171/CMH/133.
[16]	G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993.
[17]	W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.
[18]	A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965.
[19]	M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122. doi: 10.1063/1.532749.
[20]	J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.
[21]	I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957.
[22]	E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824. doi: 10.2307/2372550.
[23]	E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973.
[24]	A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123. doi: 10.1007/BF02254657.
[25]	S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151. doi: 10.1007/BF01446421.
[26]	S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893.
[27]	S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971.
[28]	S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235.
[29]	S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967.
[30]	J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456.
[31]	B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.
[32]	B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.
[33]	B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010.
[34]	P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972.
[35]	M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940. doi: 10.1103/PhysRevA.52.936.
[36]	J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111.
[37]	J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.
[38]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999.
[39]	J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884.
[40]	K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179.
[41]	K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6.
[42]	K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85. doi: 10.3792/pjaa.73.82.
[43]	K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956.
[44]	J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990.
[45]	M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.
[46]	C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001.
[47]	J.-P. Serre, Géométrie algébrique et géométrie analytique (French), Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 1-42.
[48]	J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37.
[49]	Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990.
[50]	S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165. doi: 10.1063/1.526085.
[51]	M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272. doi: 10.1109/TAC.1985.1103934.
[52]	H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988.
[53]	H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58.
[54]	H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135.
[55]	H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158.
[56]	M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003.
[57]	E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280.
[58]	E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64.
[59]	E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114.
[60]	E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.
[61]	E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85.
[62]	E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60.

show all references

References:

[1]	C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474.
[2]	C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649. doi: 10.1088/0305-4470/30/13/015.
[3]	C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614. doi: 10.1088/0305-4470/31/31/008.
[4]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010.
[5]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517.
[6]	R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991.
[7]	A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986.
[8]	J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000.
[9]	J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258.
[10]	J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87. doi: 10.1023/A:1010743114995.
[11]	J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp.
[12]	J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69. doi: 10.1023/A:1013913930134.
[13]	G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779.
[14]	G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.
[15]	G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519. doi: 10.4171/CMH/133.
[16]	G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993.
[17]	W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.
[18]	A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965.
[19]	M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122. doi: 10.1063/1.532749.
[20]	J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.
[21]	I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957.
[22]	E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824. doi: 10.2307/2372550.
[23]	E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973.
[24]	A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123. doi: 10.1007/BF02254657.
[25]	S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151. doi: 10.1007/BF01446421.
[26]	S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893.
[27]	S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971.
[28]	S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235.
[29]	S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967.
[30]	J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456.
[31]	B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.
[32]	B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.
[33]	B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010.
[34]	P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972.
[35]	M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940. doi: 10.1103/PhysRevA.52.936.
[36]	J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111.
[37]	J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.
[38]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999.
[39]	J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884.
[40]	K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179.
[41]	K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6.
[42]	K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85. doi: 10.3792/pjaa.73.82.
[43]	K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956.
[44]	J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990.
[45]	M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.
[46]	C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001.
[47]	J.-P. Serre, Géométrie algébrique et géométrie analytique (French), Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 1-42.
[48]	J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37.
[49]	Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990.
[50]	S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165. doi: 10.1063/1.526085.
[51]	M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272. doi: 10.1109/TAC.1985.1103934.
[52]	H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988.
[53]	H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58.
[54]	H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135.
[55]	H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158.
[56]	M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003.
[57]	E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280.
[58]	E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64.
[59]	E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114.
[60]	E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.
[61]	E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85.
[62]	E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60.

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